Formulation in The Complex Plane
The problem becomes more natural if we formulate it in the complex plane. We can rewrite the formula for a trigonometric polynomial as
where i is the imaginary unit. If we set z = eix, then this becomes
This reduces the problem of trigonometric interpolation to that of polynomial interpolation on the unit circle. Existence and uniqueness for trigonometric interpolation now follows immediately from the corresponding results for polynomial interpolation.
For more information on formulation of trigonometric interpolating polynomials in the complex plane see, p135 Interpolation using Fourier Polynomials.
Read more about this topic: Trigonometric Interpolation
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